ECE 5984: Power Distribution System Analysis
Lecture 2: Load Allocation across a Feeder Reference: Textbook, Chapter 2
Instructor: V. Kekatos
1
Facts
• In transmission systems, load forecasting is pretty accurate (law of large numbers) • In distribution systems, load exhibits higher variability
Motivation for allocating load
2
Problem
• Study variations of individual and aggregate loads, in particular their max values
Why?
• To size equipment (transformers) and study voltage drops
• Demand: load (kW, kVA, kVAR, A) averaged over a time period (e.g., 15 min)
Individual customer load
3
• Individual loads vary significantly; interested in the maximum
• Characterize variability for transformer sizing; load allocation; voltage drops
example: the 15-min demand ending at 6:30 pm is 4.75 kW
Average and maximum demand
4
• Collect demands across time for customer n {dn,t}Tt=1
• Average demand
• Maximum demand
example : d̃1 = 2.46 kWd̃n :=1
T
TX
t=1
dn,t<latexit sha1_base64="x5wsFQ0W8L/+Y6tVMcASJ9ZBy9I=">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</latexit><latexit sha1_base64="x5wsFQ0W8L/+Y6tVMcASJ9ZBy9I=">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</latexit><latexit sha1_base64="x5wsFQ0W8L/+Y6tVMcASJ9ZBy9I=">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</latexit><latexit sha1_base64="x5wsFQ0W8L/+Y6tVMcASJ9ZBy9I=">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</latexit>
d̄n := maxt
{dn,t}<latexit sha1_base64="rpgqjKcwJotPZ2A11WoCQRci+Ao=">AAACA3icZVBNS8QwEE39dv2qevRgcBE8LEsrgiIIghePCq4u2FLSNLsbTNKSTMWl9Li/Zm+iR3+CP8B/Y7rbg+sOhHl5M5PJe3EmuAHP+3EWFpeWV1bX1hsbm1vbO+7u3qNJc01Zh6Yi1d2YGCa4Yh3gIFg304zIWLCn+OWmqj+9Mm14qh5gmLFQkr7iPU4JWCpyD4OY6CIpI3V5FUjyFhVQ4qBIokK1oAzKyG16bW8SeB74NWiiOu4i9ztIUppLpoAKYsyz72UQFkQDp4KVjSA3LCP0hfTZs4WKSGbCYiKkxMeWSXAv1fYowBP270RBpJEEBi2bzVDGVYaBbPWZqq72gapqZpdA7yIsuMpyYIpOd/RygSHFlSE44ZpREEMLCNXcfhPTAdGEgrWtYfX7/9XOg8fTtu+1/fuz5rVfO7GGDtAROkE+OkfX6BbdoQ6iaITG6AN9OiNn7Lw7H9PWBaee2Ucz4Xz9AqoamBM=</latexit><latexit sha1_base64="rpgqjKcwJotPZ2A11WoCQRci+Ao=">AAACA3icZVBNS8QwEE39dv2qevRgcBE8LEsrgiIIghePCq4u2FLSNLsbTNKSTMWl9Li/Zm+iR3+CP8B/Y7rbg+sOhHl5M5PJe3EmuAHP+3EWFpeWV1bX1hsbm1vbO+7u3qNJc01Zh6Yi1d2YGCa4Yh3gIFg304zIWLCn+OWmqj+9Mm14qh5gmLFQkr7iPU4JWCpyD4OY6CIpI3V5FUjyFhVQ4qBIokK1oAzKyG16bW8SeB74NWiiOu4i9ztIUppLpoAKYsyz72UQFkQDp4KVjSA3LCP0hfTZs4WKSGbCYiKkxMeWSXAv1fYowBP270RBpJEEBi2bzVDGVYaBbPWZqq72gapqZpdA7yIsuMpyYIpOd/RygSHFlSE44ZpREEMLCNXcfhPTAdGEgrWtYfX7/9XOg8fTtu+1/fuz5rVfO7GGDtAROkE+OkfX6BbdoQ6iaITG6AN9OiNn7Lw7H9PWBaee2Ucz4Xz9AqoamBM=</latexit><latexit sha1_base64="rpgqjKcwJotPZ2A11WoCQRci+Ao=">AAACA3icZVBNS8QwEE39dv2qevRgcBE8LEsrgiIIghePCq4u2FLSNLsbTNKSTMWl9Li/Zm+iR3+CP8B/Y7rbg+sOhHl5M5PJe3EmuAHP+3EWFpeWV1bX1hsbm1vbO+7u3qNJc01Zh6Yi1d2YGCa4Yh3gIFg304zIWLCn+OWmqj+9Mm14qh5gmLFQkr7iPU4JWCpyD4OY6CIpI3V5FUjyFhVQ4qBIokK1oAzKyG16bW8SeB74NWiiOu4i9ztIUppLpoAKYsyz72UQFkQDp4KVjSA3LCP0hfTZs4WKSGbCYiKkxMeWSXAv1fYowBP270RBpJEEBi2bzVDGVYaBbPWZqq72gapqZpdA7yIsuMpyYIpOd/RygSHFlSE44ZpREEMLCNXcfhPTAdGEgrWtYfX7/9XOg8fTtu+1/fuz5rVfO7GGDtAROkE+OkfX6BbdoQ6iaITG6AN9OiNn7Lw7H9PWBaee2Ucz4Xz9AqoamBM=</latexit><latexit sha1_base64="rpgqjKcwJotPZ2A11WoCQRci+Ao=">AAACA3icZVBNS8QwEE39dv2qevRgcBE8LEsrgiIIghePCq4u2FLSNLsbTNKSTMWl9Li/Zm+iR3+CP8B/Y7rbg+sOhHl5M5PJe3EmuAHP+3EWFpeWV1bX1hsbm1vbO+7u3qNJc01Zh6Yi1d2YGCa4Yh3gIFg304zIWLCn+OWmqj+9Mm14qh5gmLFQkr7iPU4JWCpyD4OY6CIpI3V5FUjyFhVQ4qBIokK1oAzKyG16bW8SeB74NWiiOu4i9ztIUppLpoAKYsyz72UQFkQDp4KVjSA3LCP0hfTZs4WKSGbCYiKkxMeWSXAv1fYowBP270RBpJEEBi2bzVDGVYaBbPWZqq72gapqZpdA7yIsuMpyYIpOd/RygSHFlSE44ZpREEMLCNXcfhPTAdGEgrWtYfX7/9XOg8fTtu+1/fuz5rVfO7GGDtAROkE+OkfX6BbdoQ6iaITG6AN9OiNn7Lw7H9PWBaee2Ucz4Xz9AqoamBM=</latexit>
example : d̄1 = 6.19 kW at 13:15<latexit sha1_base64="eEVtCYxHrDv6YiEGW0eP2HizzUA=">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</latexit><latexit sha1_base64="eEVtCYxHrDv6YiEGW0eP2HizzUA=">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</latexit><latexit sha1_base64="eEVtCYxHrDv6YiEGW0eP2HizzUA=">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</latexit><latexit sha1_base64="eEVtCYxHrDv6YiEGW0eP2HizzUA=">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</latexit>
• How to calculate total energy consumed?
Load factor and demand factor
5
• Load factor LFn :=d̃nd̄n
1 [why?]
example : LF1 =d̃1d̄1
=2.46 kW
6.19 kW= 0.40
LF=1 is ideal for better utilization of facilities
• Demand factor
Demand factor :=maximum customer demand
sum of device ratings=
6.19 kW
35 kW= 0.18 1
percentage of electrical devices that are on when maximum demand occurs
Characterized load for one customer. How about aggregated load in a distribution transformer serving 5-50 customers?
Load surveys
6
• We are metering energy monthly (billing); but do not know user maximum
• Interested in maxima to determine transformer ratings
• Install meters over some customers and use linear regression to find a linear transformation between energy consumption and maximum demand
d̄n ' ↵ · d̃n + �<latexit sha1_base64="EyAmX+IV+J30jNxMeVqe3VL+poA=">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</latexit><latexit sha1_base64="EyAmX+IV+J30jNxMeVqe3VL+poA=">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</latexit><latexit sha1_base64="EyAmX+IV+J30jNxMeVqe3VL+poA=">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</latexit><latexit sha1_base64="EyAmX+IV+J30jNxMeVqe3VL+poA=">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</latexit>
Load at distribution transformer
7
Customer #1
Customer #2
Customer #3
Customer #4 Diversified demand: sum of customer demands
d =NX
n=1
dn
Load duration curve
8
• Sort diversified demand to evaluate transformer stress
• Each bar corresponds to 15min/24h=1.04%
• Example: 22% of the time, the transformer serves more than 12 kW
Maximum demand for load aggregations
9
• Maximum diversified demand (max of sum)
d̄ := maxt
{dt} = maxt
(NX
n=1
dn,t
)
• Maximum non-coincident diversified demand (sum of max)
d̂ :=NX
n=1
d̄n =NX
n=1
maxt
{dn,t} example : d̂ = 6.18 + 6.82 + 4.93 + 7.05 = 24.98 kW
example : d̄ = 16.16 kW at 17:30
Per-customer maxima do not necessarily occur at the same time …
• Diversity factor DF :=d̂
d̄� 1 [why?] example : DF =
d̂
d̄= 1.55
• Load diversity example : 24.98� 16.16 = 8.82 kW
Why do we care about the max non-coincident diversified demand?
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Diversity factors
10
• Maximum diversified demand is hard to measure
• We usually know max demand per customer (metered or via linear regression from monthly energy)
§ record at specific network locations § record for all customers downstream these locations
• How to translate from d̂ ! d̄?
• Calculate diversity factors experimentally
d̄
{d̄n}
DFN
N
Recap
11
1. For each customer, either directly meter max demand , or use linear regression from customer’s total demand
2. Sum per-customer maxima to find
3. Use experimentally estimated DFN to find max diversified demand
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d̃n<latexit sha1_base64="b8ovXiyISZXm+r08DQ+dGBuV8Fs=">AAAB7XicZVBNSwMxEM36WetX1aOXxSJ4KGVXBD0WvHisYD+gXUo2O9uGZpMlmRXK0p/Rm9Sjv8Yf4L8x2+7B2gdhXt5kMvMmTAU36Hk/zs7u3v7BYeWoenxyenZeu7jsGpVpBh2mhNL9kBoQXEIHOQropxpoEgrohdPnIt97B224km84SyFI6FjymDOKVhoMkYsI8mg+kqNa3Wt6K7jbxC9JnZRoj2rfw0ixLAGJTFBjBr6XYpBTjZwJmFeHmYGUsikdw8BSSRMwQb4aee7eWiVyY6Xtkeiu1L8VOU1MQnHSsNHMkrCIOEkaY5DF1X5QZM1mE4yfgpzLNEOQbN0jzoSLyi2suxHXwFDMLKFMczumyyZUU4Z2QVXr3//vdpt075u+1/RfH+otv9xEhVyTG3JHfPJIWuSFtEmHMKLIgizJp6OchfPhLNdPd5yy5opswPn6BUQYj3o=</latexit><latexit sha1_base64="b8ovXiyISZXm+r08DQ+dGBuV8Fs=">AAAB7XicZVBNSwMxEM36WetX1aOXxSJ4KGVXBD0WvHisYD+gXUo2O9uGZpMlmRXK0p/Rm9Sjv8Yf4L8x2+7B2gdhXt5kMvMmTAU36Hk/zs7u3v7BYeWoenxyenZeu7jsGpVpBh2mhNL9kBoQXEIHOQropxpoEgrohdPnIt97B224km84SyFI6FjymDOKVhoMkYsI8mg+kqNa3Wt6K7jbxC9JnZRoj2rfw0ixLAGJTFBjBr6XYpBTjZwJmFeHmYGUsikdw8BSSRMwQb4aee7eWiVyY6Xtkeiu1L8VOU1MQnHSsNHMkrCIOEkaY5DF1X5QZM1mE4yfgpzLNEOQbN0jzoSLyi2suxHXwFDMLKFMczumyyZUU4Z2QVXr3//vdpt075u+1/RfH+otv9xEhVyTG3JHfPJIWuSFtEmHMKLIgizJp6OchfPhLNdPd5yy5opswPn6BUQYj3o=</latexit><latexit sha1_base64="b8ovXiyISZXm+r08DQ+dGBuV8Fs=">AAAB7XicZVBNSwMxEM36WetX1aOXxSJ4KGVXBD0WvHisYD+gXUo2O9uGZpMlmRXK0p/Rm9Sjv8Yf4L8x2+7B2gdhXt5kMvMmTAU36Hk/zs7u3v7BYeWoenxyenZeu7jsGpVpBh2mhNL9kBoQXEIHOQropxpoEgrohdPnIt97B224km84SyFI6FjymDOKVhoMkYsI8mg+kqNa3Wt6K7jbxC9JnZRoj2rfw0ixLAGJTFBjBr6XYpBTjZwJmFeHmYGUsikdw8BSSRMwQb4aee7eWiVyY6Xtkeiu1L8VOU1MQnHSsNHMkrCIOEkaY5DF1X5QZM1mE4yfgpzLNEOQbN0jzoSLyi2suxHXwFDMLKFMczumyyZUU4Z2QVXr3//vdpt075u+1/RfH+otv9xEhVyTG3JHfPJIWuSFtEmHMKLIgizJp6OchfPhLNdPd5yy5opswPn6BUQYj3o=</latexit><latexit sha1_base64="b8ovXiyISZXm+r08DQ+dGBuV8Fs=">AAAB7XicZVBNSwMxEM36WetX1aOXxSJ4KGVXBD0WvHisYD+gXUo2O9uGZpMlmRXK0p/Rm9Sjv8Yf4L8x2+7B2gdhXt5kMvMmTAU36Hk/zs7u3v7BYeWoenxyenZeu7jsGpVpBh2mhNL9kBoQXEIHOQropxpoEgrohdPnIt97B224km84SyFI6FjymDOKVhoMkYsI8mg+kqNa3Wt6K7jbxC9JnZRoj2rfw0ixLAGJTFBjBr6XYpBTjZwJmFeHmYGUsikdw8BSSRMwQb4aee7eWiVyY6Xtkeiu1L8VOU1MQnHSsNHMkrCIOEkaY5DF1X5QZM1mE4yfgpzLNEOQbN0jzoSLyi2suxHXwFDMLKFMczumyyZUU4Z2QVXr3//vdpt075u+1/RfH+otv9xEhVyTG3JHfPJIWuSFtEmHMKLIgizJp6OchfPhLNdPd5yy5opswPn6BUQYj3o=</latexit>
d̂N =NX
n=1
d̄n<latexit sha1_base64="B/OBTkIwCMWNwOENqsjcMLfQCoQ=">AAACA3icZVDLSsNAFJ3UV62vqksXBovgopREBN0UCm5cSQX7gCaWyXTSDJ2ZhJkboYQu+zXdSV36CX6Af+Ok7cLaC8M999zX3BMknGlwnB+rsLW9s7tX3C8dHB4dn5RPz9o6ThWhLRLzWHUDrClnkraAAafdRFEsAk47wegxz3feqdIslq8wTqgv8FCykBEMhuqXL70IQzaYvD3XPZ2Kfibrrgm8ACvD9k1Fxak5C7M3gbsCFbSyZr/87Q1ikgoqgXCsdc91EvAzrIARTiclL9U0wWSEh7RnoMSCaj9bHDKxrw0zsMNYmSfBXrB/OzIstMAQVY3XYxHkHiJRHVKZh2ZAntXrSyB88DMmkxSoJMsdYcptiO1cEHvAFCXAxwZgopj5pk0irDABI1vJ3O/+v3YTtG9rrlNzX+4qDXelRBFdoCt0g1x0jxroCTVRCxE0RTM0R5/W1JpZH9Z8WVqwVj3naM2sr1+TT5gE</latexit><latexit sha1_base64="B/OBTkIwCMWNwOENqsjcMLfQCoQ=">AAACA3icZVDLSsNAFJ3UV62vqksXBovgopREBN0UCm5cSQX7gCaWyXTSDJ2ZhJkboYQu+zXdSV36CX6Af+Ok7cLaC8M999zX3BMknGlwnB+rsLW9s7tX3C8dHB4dn5RPz9o6ThWhLRLzWHUDrClnkraAAafdRFEsAk47wegxz3feqdIslq8wTqgv8FCykBEMhuqXL70IQzaYvD3XPZ2Kfibrrgm8ACvD9k1Fxak5C7M3gbsCFbSyZr/87Q1ikgoqgXCsdc91EvAzrIARTiclL9U0wWSEh7RnoMSCaj9bHDKxrw0zsMNYmSfBXrB/OzIstMAQVY3XYxHkHiJRHVKZh2ZAntXrSyB88DMmkxSoJMsdYcptiO1cEHvAFCXAxwZgopj5pk0irDABI1vJ3O/+v3YTtG9rrlNzX+4qDXelRBFdoCt0g1x0jxroCTVRCxE0RTM0R5/W1JpZH9Z8WVqwVj3naM2sr1+TT5gE</latexit><latexit sha1_base64="B/OBTkIwCMWNwOENqsjcMLfQCoQ=">AAACA3icZVDLSsNAFJ3UV62vqksXBovgopREBN0UCm5cSQX7gCaWyXTSDJ2ZhJkboYQu+zXdSV36CX6Af+Ok7cLaC8M999zX3BMknGlwnB+rsLW9s7tX3C8dHB4dn5RPz9o6ThWhLRLzWHUDrClnkraAAafdRFEsAk47wegxz3feqdIslq8wTqgv8FCykBEMhuqXL70IQzaYvD3XPZ2Kfibrrgm8ACvD9k1Fxak5C7M3gbsCFbSyZr/87Q1ikgoqgXCsdc91EvAzrIARTiclL9U0wWSEh7RnoMSCaj9bHDKxrw0zsMNYmSfBXrB/OzIstMAQVY3XYxHkHiJRHVKZh2ZAntXrSyB88DMmkxSoJMsdYcptiO1cEHvAFCXAxwZgopj5pk0irDABI1vJ3O/+v3YTtG9rrlNzX+4qDXelRBFdoCt0g1x0jxroCTVRCxE0RTM0R5/W1JpZH9Z8WVqwVj3naM2sr1+TT5gE</latexit><latexit sha1_base64="B/OBTkIwCMWNwOENqsjcMLfQCoQ=">AAACA3icZVDLSsNAFJ3UV62vqksXBovgopREBN0UCm5cSQX7gCaWyXTSDJ2ZhJkboYQu+zXdSV36CX6Af+Ok7cLaC8M999zX3BMknGlwnB+rsLW9s7tX3C8dHB4dn5RPz9o6ThWhLRLzWHUDrClnkraAAafdRFEsAk47wegxz3feqdIslq8wTqgv8FCykBEMhuqXL70IQzaYvD3XPZ2Kfibrrgm8ACvD9k1Fxak5C7M3gbsCFbSyZr/87Q1ikgoqgXCsdc91EvAzrIARTiclL9U0wWSEh7RnoMSCaj9bHDKxrw0zsMNYmSfBXrB/OzIstMAQVY3XYxHkHiJRHVKZh2ZAntXrSyB88DMmkxSoJMsdYcptiO1cEHvAFCXAxwZgopj5pk0irDABI1vJ3O/+v3YTtG9rrlNzX+4qDXelRBFdoCt0g1x0jxroCTVRCxE0RTM0R5/W1JpZH9Z8WVqwVj3naM2sr1+TT5gE</latexit>
d̄N =d̂N
DFN<latexit sha1_base64="zQ9ZxLKpM6CL56rRDawxo7FwU88=">AAACC3icZVDLSsNAFJ3UV62vqks3wSK6KCURQTeCoIirUsE+wNQymd60g5NJmLkRS8gn9Gu6k7p06Qf4N07aLnxcGO65586dO+f4seAaHefLKiwtr6yuFddLG5tb2zvl3b2WjhLFoMkiEamOTzUILqGJHAV0YgU09AW0/efrvN9+AaV5JB9wFEM3pAPJA84oGqpXPvZ8qtJ+9lS/9AJFWeoNKc7qLPUQXjG9uc169axXrjg1Zxb2f+AuQIUsotErf3r9iCUhSGSCav3oOjF2U6qQMwFZyUs0xJQ90wE8GihpCLqbzgRl9pFh+nYQKXMk2jP250RKQx1SHFZN1qPQzzMOw+oAZF6aB/Ku/r0Eg4tuymWcIEg23xEkwsbIzo2x+1wBQzEygDLFzTdtNqTGEjT2lYx+96/a/6B1WnOdmnt/VrlyF04UyQE5JCfEJefkityRBmkSRsZkQqbk3RpbE+vNms6vFqzFzD75FdbHN3cWm9s=</latexit><latexit sha1_base64="zQ9ZxLKpM6CL56rRDawxo7FwU88=">AAACC3icZVDLSsNAFJ3UV62vqks3wSK6KCURQTeCoIirUsE+wNQymd60g5NJmLkRS8gn9Gu6k7p06Qf4N07aLnxcGO65586dO+f4seAaHefLKiwtr6yuFddLG5tb2zvl3b2WjhLFoMkiEamOTzUILqGJHAV0YgU09AW0/efrvN9+AaV5JB9wFEM3pAPJA84oGqpXPvZ8qtJ+9lS/9AJFWeoNKc7qLPUQXjG9uc169axXrjg1Zxb2f+AuQIUsotErf3r9iCUhSGSCav3oOjF2U6qQMwFZyUs0xJQ90wE8GihpCLqbzgRl9pFh+nYQKXMk2jP250RKQx1SHFZN1qPQzzMOw+oAZF6aB/Ku/r0Eg4tuymWcIEg23xEkwsbIzo2x+1wBQzEygDLFzTdtNqTGEjT2lYx+96/a/6B1WnOdmnt/VrlyF04UyQE5JCfEJefkityRBmkSRsZkQqbk3RpbE+vNms6vFqzFzD75FdbHN3cWm9s=</latexit><latexit sha1_base64="zQ9ZxLKpM6CL56rRDawxo7FwU88=">AAACC3icZVDLSsNAFJ3UV62vqks3wSK6KCURQTeCoIirUsE+wNQymd60g5NJmLkRS8gn9Gu6k7p06Qf4N07aLnxcGO65586dO+f4seAaHefLKiwtr6yuFddLG5tb2zvl3b2WjhLFoMkiEamOTzUILqGJHAV0YgU09AW0/efrvN9+AaV5JB9wFEM3pAPJA84oGqpXPvZ8qtJ+9lS/9AJFWeoNKc7qLPUQXjG9uc169axXrjg1Zxb2f+AuQIUsotErf3r9iCUhSGSCav3oOjF2U6qQMwFZyUs0xJQ90wE8GihpCLqbzgRl9pFh+nYQKXMk2jP250RKQx1SHFZN1qPQzzMOw+oAZF6aB/Ku/r0Eg4tuymWcIEg23xEkwsbIzo2x+1wBQzEygDLFzTdtNqTGEjT2lYx+96/a/6B1WnOdmnt/VrlyF04UyQE5JCfEJefkityRBmkSRsZkQqbk3RpbE+vNms6vFqzFzD75FdbHN3cWm9s=</latexit><latexit sha1_base64="zQ9ZxLKpM6CL56rRDawxo7FwU88=">AAACC3icZVDLSsNAFJ3UV62vqks3wSK6KCURQTeCoIirUsE+wNQymd60g5NJmLkRS8gn9Gu6k7p06Qf4N07aLnxcGO65586dO+f4seAaHefLKiwtr6yuFddLG5tb2zvl3b2WjhLFoMkiEamOTzUILqGJHAV0YgU09AW0/efrvN9+AaV5JB9wFEM3pAPJA84oGqpXPvZ8qtJ+9lS/9AJFWeoNKc7qLPUQXjG9uc169axXrjg1Zxb2f+AuQIUsotErf3r9iCUhSGSCav3oOjF2U6qQMwFZyUs0xJQ90wE8GihpCLqbzgRl9pFh+nYQKXMk2jP250RKQx1SHFZN1qPQzzMOw+oAZF6aB/Ku/r0Eg4tuymWcIEg23xEkwsbIzo2x+1wBQzEygDLFzTdtNqTGEjT2lYx+96/a/6B1WnOdmnt/VrlyF04UyQE5JCfEJefkityRBmkSRsZkQqbk3RpbE+vNms6vFqzFzD75FdbHN3cWm9s=</latexit>
Load allocation using diversity factors
12
Example 2.1: Analyze a single-phase lateral. Given: a) monthly energy usage per customer; assume 0.9 PF lagging
b) load survey shows that
c) diversity factors (see graph on previous slide) Wanted: max diversified demand for each transformer and line segment
d̄n = 0.2 + 0.008 ·Monthly energy [kWh]
33.7 kVA 39.4 kVA 54.4 kVA
54.4 kVA80.77 kVA103.22 kVA
KCL load does not apply with max aggregated (real or apparent) powers!
Transformer ratings: T1: 25 kVA T2: 37.5 kVA T3: 50 kVA
Voltage drop using diversity factors
13
Example: Find voltage at each transformer assuming VN1=2.4 kV
33.7 kVA 39.4 kVA 54.4 kVA
54.4 kVA80.77 kVA103.22 kVA
zline = 0.3 + j0.6 ⌦/mile
5, 000 ft 500 ft 750 ft
1 mile = 5, 280 ft
232.15 V [3.27%] 233.1 V [2.92%] 232.7 V [3.04%]
T1 : 25 kVA, 2400� 240 V, Z = 1.8\40�%T2 : 37.5 kVA, 2400� 240 V, Z = 1.9\45�%T3 : 50 kVA, 2400� 240 V, Z = 2.0\50�%
Transformer load management
14
d1) DFN’s for different N d2) customer maxima; and d3) customer assignment to transformers
• Training stage
- Collect historical data from transformers
- Fit a linear regression model fitting monthly energy served to peak demand
• Operational stage
- Knowing d3) and customer bills, find total energy to be served
- Using the trained model, predict max demand
• Transformer load management is a simpler but less accurate method – relies on the fact that transformers are metered in greater detail
• The method of diversity factors requires knowing
Allocation factors
15
Much easier to implement; more conservative; but relies on reliable ratings
Distribute maximum feeder demand on transformers based on their rating
allocation factor d̄Tk = d̄feeder ⇥kVATkPm kVATm
Example: Re-allocate load assuming maximum demand at feeder is 92.9 kW
22.94 kVA 34.41 kVA 45.88 kVA
Voltage drop using allocation factors
16
Example: Find voltage at each transformer using allocation factors
• Due to linearity of allocation factors, KCL now holds for line segments
Does this method yield the actual quantities assuming demands are correct?
92.9 + j45 kVA 72.3 + j35 kVA
31 + j15 kVA21 + j10 kVA
41.3 + j20 kVA
41.3 + j20 kVA
233.5\� 0.9� V233.7\� 0.8� V234.0\� 0.6� V
• Results obtained are similar; this method may be less conservative
• Note small voltage angles
Feeder load
17
• Smoother demand curve
• Define statistics similar to transformers (maximum diversified demand)
Summary
18
• Allocate load to size transformer and study voltage drops
• Method 1: diversity factors
• Method 2: transformer load management
• Method 3: allocation factors d̄Tk =kVATkPm kVATm
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d̃n �!X
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regressiond̄Tk
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d̃nlinear�!
regressiond̄n �! d̂Tk =
X
n2Tk
d̄nDFN�! d̄Tk
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• Requirements for every method
• M3 features linearity in flows; none of the methods considers losses